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Coarse-graining and Entropy Production in a Climate Model

- Part 3-

Valerio LucariniKlimacampus, Meteorological Institute, University of Hamburg

Department of Mathematics & Statistics, University of Reading

Email: valerio.lucarini@uni-hamburg.de

Cambridge, 20/11/2013

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TodayInterplay between order and chaos in the

climate systemFrom space, mid-latitude cyclones look like

Von Karman’s vorticesBut atmosphere is NOT 2D at all

Dynamical/thermodynamical processesWe’ll focus on entropy production and

coarse graining: irreversibility at various scales! How to account for them?

EP dissipation parametrizations

Scales of Motions (Smagorinsky)

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Update

ff

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Atmospheric MotionsThree contrasting approaches:

Those who like maps, look for features/particlesThose who like regularity, look for wavesThose who like irreversibility, look for

turbulence

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CommentsNone of the 3 approaches is fully satisfactory

The atmosphere has indeed features, but organised motions are just part of the story

Particles have a life-cycle: what generates them, what dissipates them?

The waves we observe result from unstable processes – energy budgets are crucial

Nonlinearities turbulent mixing and dissipationScaling properties are observed at larger scales:

turbulence is macro-turbulenceBaroclinic (3D) / barotropic (2D) interplay Lorenz Energy Cycle

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Other peculiarities of the atmosphereNegative viscosity phenomena:

Turbulent momentum fluxes can be directed towards the region of high momentum (Jets)

Turbulent: fast, non-zonally symmetric (but far from microscopic, infinitesimal!)

Discovered by V. StarrThe atmosphere is not a “simple”, Onsager-

like dissipative system – eddy viscosity?

Such phenomena are due to long-range correlations (travelling waves!)

… but how can this be sustained?

Disequilibrium in the Earth system

(Kleidon, 2011)

climate

Disequilibrium

Work

Irreversibility

Multiscale

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Main Formulas for Climate Thermodynamics

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Entropy ProductionContributions of dissipation plus heat

transport:

Note:If heat transport along T is strong, η is smallIf the transport along T is weak, α is small

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But…Two ways to compute EP:

Direct vs Indirect

Material vs Radiative

T R A N S P O R T

E N E R G Y

E N E R G Y

T R A N S P O R T

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Transport, Mixing

Vertical Transport of Energy Convective adjustmentIrreversible mixing

Horizontal TransportsBaroclinic adjustmentIrreversible mixing

W

C

W C

2-box model(s)

4-box model of entropy budget

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1

3 4

2

Poleward transport

Vertical transport

Fluid

Surface

2×2 box model

Results on IPCC GCMs

Hor vs Vert EP in IPCC models

Warmer climate:Hor↓ Vert↑

Venus, Mars, Titan15

vertinS

horinS

L., Ragone, Fraedrich, 2011

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A step backwards

Let’s consider a quasi-equilibrium system where Onsager relations apply

with A positive definite quadratic formEP is locally positive!Let’s use extensively the Parseval

Theorem for

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We obtain:

Where is the entropy produced at space scale kl and t scale ωm

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Coarse GrainingWe perform a coarse graining on our data

cut-off for |ωm| > |ωMAX| and |kl |> |kMAX|

Estimate of EP:

And: the coarser the graining (lower values for the cut-off), the lower the value of the estimate of the EP

Each scale contributes with a positive termDifferent scales tell about different processes

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Local MadnessIn the stratosphere

eddy (high frequency/wavenumber) heat fluxes are locally against the T gradient

NEGATIVE EPHow it possible?Work on the systemLocal vs Global?

Scales?

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Entropy Production in the CS

It is NOT a diffusive system, as we have seen

It is a non-equilibrium, multiphase system

What are our expectations on the properties of ?

Climate is VERY heterogeneous (horizontal vs vertical), and seasonally and daily forced

Feedbacks, re-equilibration processes

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Computing entropy production

Direct Formula

Indirect Formula

We use high resolution fields

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Coarse graining the fieldsWe perform coarse graining with a given

temporal kernel τ and spatial kernel v

Coarse grained estimates

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Temporal Coarse Graining

Time averaging kernel

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Time and Space Coarse GrainingAveraging in time and along the pressure levels

Direct Method Indirect Method

1 D COLUMN 1 D COLUMN

CL

IMA

TO

LO

GY

CL

IMA

TO

LO

GY

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EP – 1 d resolution – Indirect MethodWe use the indirect formula

Ver Av

Hor

Av

1 D COLUMN

2 D

HO

R F

LO

W

0 D

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EP - 1d resolution – Direct MethodSpatial coarse graining does not kill EP!

Ver Av

Hor

Av

1 D COLUMN

2 D

HO

R F

LO

W

0 D

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Transport, Mixing – Indirect FormulaHorizontal Transports

Baroclinic adjustmentIrreversible mixing 6 mWK-1m-2

W C

2-box model

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Transport, Mixing – Direct FormulaHorizontal Transports

Hydrological cycleDissipation of KE18 mWK-1m-2

W C

2-box model

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CommentsFull Coarse graining

Indirect formula gives 0: the system is reduced to a 0D body, which absorbs and emits radiation at a given temperature

Direct formula gives min EP = SKE

Concluding we get: 13 mWK-1m-2 SKE

6 mWK-1m-2 Shor

33 mWK-1m-2 Sver

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Comment The coarser the graining, the lower the

estimated value of entropy productionWe lose information about transfer of energy

across domains at different temperatures

From indirect method: can separate vertical/ horizontal only transfer processesLosing info on the vertical structure is very badremoving all resolution gives 0

Direct Method: we retain the effect of KE dissipation also at lowest resolution

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EP 1d res: Direct-Indirect MethodDifference depends on effective resolutionDifference >0, 0 with full resolution

NVER

Hor

Av

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InterestingLet’s compare the effect of coarse graining

Note that

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InterpretationSystem is forced, fluctuates and

dissipatesThe correlation btw radiative heating

rates & temperature >0 at all scales Energy input temperature increase

The correlation btw material heating rates & temperature <0 at all scales T fluctuation relaxation

System driven by radiative forcing, T fluctuations dissipated by material fluxes

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What we have learnedPositive contribution to EP comes from

all time and space scalesNot so locally! Waves allow for this.13 mWK-1m-2 SKE

6 mWK-1m-2 Shor

33 mWK-1m-2 Sver

This looks like being a general resultIt may be a way to generalize Onsager-like

properties to far from equilibrium systems

Should be used in other, simpler systems as well (Aquaplanet, Rayleigh-Benard)

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Bibliography Lucarini V. and S. Pascale, Entropy Production and Coarse Graining of the

Climate Fields in a General Circulation Model, submitted to Clim. Dyn. (2013)

Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the habitable zone: a thermodynamic investigation, Icarus (2013)

Johnson D.R., Entropy, the Lorenz Energy Cycle and Climate, 659-720 in General Circulation Model Development: Past, Present and Future, D.A. Randall Ed. (Academic Press, 2002)

Kleidon, A., Lorenz, R.D. (Eds.) Non-equilibrium thermodynamics and the production of entropy: life, Earth, and beyond (Springer, 2005)

Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate System, Phys Rev. E 80, 021118 (2009)

Lucarini V., Modeling Complexity: the case of Climate Science, in “Models, Simulations, and the Reduction of Complexity”, Gähde U, Hartmann S, Wolf J. H., De Gruyter Eds., Hamburg (2013)

Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on the thermodynamical properties of the climate system. J. Atmos. Sci., 68, 2438-2458

Saltzman B., Dynamic Paleoclimatology (Academic Press, 2002)